Question

help please, bottom text, bottom text, bottom text

Answer 1

We have the following functions:

$\sf \begin{cases} \sf f(x) = - 2x {}^{2} - 2 \\ \sf g(x) = - 3x + 2 \\ \sf h(x) = - 3 \end{cases}$

The question asks us what is the result of the composite function formed by:

$\sf f(g(f(h(0))))$

To find the answer, we must solve from the inside out, that is, we must start the h (x) function:

$\sf f(g(f( - 3)))$

To calculate f (-3), you simply substitute this value of "x" in the function f (x):

$\sf f( - 3) = - 2.( - 3) {}^{2} - 2 \\ \sf f( - 3) = - 2.9 - 2 \: \: \: \: \: \: \: \: \: \\ \sf f( - 3 ) = - 18 - 2 \: \: \: \: \: \: \: \: \: \: \\ \sf f( - 3) = - 20 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Replacing the result in the remaining functions:

$\sf f(g( - 20))$

Now just follow the same logic as the previous calculation:

$\sf g( - 20) = - 3.( - 20) + 2 \\ \sf g( - 20 ) = 60 + 2 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \sf g( - 20) = 62 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Replacing:

$\sf f(62) = - 2x {}^{2} - 2 \: \: \: \: \: \: \: \: \: \: \\ \sf f(62) = - 2.( 62) {}^{2} - 2 \: \: \: \\ \sf f(62) = -2.(3844) - 2 \\ \sf f(62) = - 7688 - 2 \: \: \: \: \: \: \: \\ \sf f(62) = - 7690 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Therefore, we can conclude that:

$\boxed{ \sf f(g(f(h(0)))) = - 7690}$

Hope this helps